Relations And Functions

Question
CBSEENMA11015538

If, for a positive integer n, the quadratic equation,
x(x + 1) + (x + 1) (x + 2) + .....
+ (x + n -1 ) (x + n) = 10n
has two consecutive integral solutions, then n is equal to :

  • 11

  • 12

  • 9

  • 10

Solution

A.

11

nx squared space plus space straight x left parenthesis 1 space plus space 3 plus 5 plus..... space plus space left parenthesis 2 straight n minus 1 right parenthesis right parenthesis space plus
space left parenthesis 1.2 space plus space 2.3 space plus space..... space plus space left parenthesis straight n minus 1 right parenthesis. straight n right parenthesis minus 10 straight n space equals space 0
rightwards double arrow space nx squared space plus space straight x space left parenthesis straight n squared right parenthesis space plus space fraction numerator straight n left parenthesis straight n squared minus 1 right parenthesis over denominator 3 end fraction space minus space 10 straight n space equals space 0
rightwards double arrow space straight x squared space plus space straight x space left parenthesis straight n right parenthesis space plus space fraction numerator left parenthesis straight n squared minus 1 right parenthesis over denominator 3 end fraction minus 10 space equals space 0 space less than subscript straight beta superscript straight alpha
left parenthesis straight alpha space minus space straight beta right parenthesis squared space equals space 1
rightwards double arrow space left parenthesis straight alpha space plus space straight beta right parenthesis squared space minus space 4 αβ space equals space 1
rightwards double arrow space straight n squared space minus space 4 space open parentheses fraction numerator straight n squared minus 1 over denominator 3 end fraction minus 10 close parentheses space equals space 1
rightwards double arrow space straight n space equals space 11

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